AMC10 2007 B

AMC10 2007 B · Q14

AMC10 2007 B · Q14. It mainly tests Systems of equations, Percent.

Some boys and girls are having a car wash to raise money for a class trip to China. Initially 40% of the group are girls. Shortly thereafter two girls leave and two boys arrive, and then 30% of the group are girls. How many girls were initially in the group?

一些男孩和女孩正在为班级去中国的旅行筹款洗车。最初小组的40%是女孩。不久之后，两个女孩离开，两个男孩到来，然后小组的30%是女孩。最初小组中有多少女孩？

(A) 4 4

(B) 6 6

(D) 10 10

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let $g$ be the number of girls and $b$ the number of boys initially in the group. Then $g = 0.4(g + b)$. After two girls leave and two boys arrive, the size of the entire group is unchanged, so $g - 2 = 0.3(g + b)$. The solution of the system of equations $g = 0.4(g + b)$ and $g - 2 = 0.3(g + b)$ is $g = 8$ and $b = 12$, so there were initially 8 girls. OR After two girls leave and two boys arrive, the size of the group is unchanged. So the two girls who left represent $40\% - 30\% = 10\%$ of the group. Thus the size of the group is 20, and the original number of girls was $40\%$ of 20, or 8.

答案（C）：设最初小组中女生人数为 $g$，男生人数为 $b$。则 $g = 0.4(g + b)$。两名女生离开、两名男生加入后，整个小组人数不变，所以 $g - 2 = 0.3(g + b)$。方程组 $g = 0.4(g + b)$ 和 $g - 2 = 0.3(g + b)$ 的解为 $g = 8$、$b = 12$，因此最初有 8 名女生。或者两名女生离开、两名男生加入后，小组总人数不变。因此离开的两名女生占小组的 $40\% - 30\% = 10\%$。所以小组人数为 20，原来的女生人数是 20 的 $40\%$，即 8。

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